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On the Witt Vector Frobenius ON THE WITT VECTOR FROBENIUS CHRISTOPHER DAVIS AND KIRAN S. KEDLAYA Abstract. We study the kernel and cokernel of the Frobenius map on the p-typical Witt vectors of a commutative ring, not necessarily of characteristic p. We give many equivalent conditions to surjectivity of the Frobenus map on both finite and infinite length Witt vectors. In particular, surjectivity on finite Witt vectors turns out to be stable under certain integral extensions; this provides a clean formulation of a strong generalization of Faltings's almost purity theorem from p-adic Hodge theory, incorporating recent improvements by Kedlaya{Liu and by Scholze. Introduction Fix a prime number p. To each ring R (always assumed commutative and with unit), we may associate in a functorial manner the ring of p-typical Witt vectors over R, denoted W (R), and an endomorphism F of W (R) called the Frobenius endomorphism. The ring W (R) is set-theoretically an infinite product of copies of R, but with an exotic ring structure when p is not a unit in R; for example, for R a perfect ring of characteristic p, W (R) is the unique strict p-ring with ∼ W (R)=pW (R) = R. In particular, for R = Fp, W (R) = Zp. In this paper, we study the kernel and cokernel of the Frobenius endomorphism on W (R). In case p = 0 in R, this map is induced by functoriality from the Frobenius endomorphism of R, and in particular is injective when R is reduced and bijective when R is perfect. If p 6= 0 in R, the Frobenius map is somewhat more mysterious. To begin with, it is never injective; it is easy to construct many elements of the kernel. On the other hand, Frobenius is surjective in some cases, although these seem to be somewhat artificial; the simplest nontrivial example we have found is the valuation subring of a spherical completion of Qp. While surjectivity of Frobenius on full Witt vectors is rather rare, some weaker conditions turn out to be more relevant to applications. For instance, one can view the full ring of Witt vectors as an inverse limit of finite-length truncations, and surjectivity of Frobenius on finite levels is satisfied quite often. For instance, this holds for R equal to the ring of integers in any infinite algebraic extension of Q which is sufficiently ramified at p (e.g., the p-cyclotomic extension). In fact, this condition can be used to give a purely ring-theoretic formulation of a very strong generalization of Faltings's almost purity theorem [2]. The theorem of Faltings features prominently in the theory of comparison isomorphisms in p-adic Hodge Date: October 1, 2012. 2010 Mathematics Subject Classification. Primary 13F35. Davis was supported by the Max-Planck-Institut f¨urMathematik (Bonn). Kedlaya was sup- ported by NSF (CAREER grant DMS-0545904, grant DMS-1101343), DARPA (grant HR0011- 09-1-0048), MIT (NEC Fund, Green professorship), UCSD (Warschawski professorship). 1 2 CHRISTOPHER DAVIS AND KIRAN S. KEDLAYA theory; the generalization in question emerged recently from work of the second author and Liu [9] and of Scholze [11]. One principal motivation for studying the Frobenius on Witt vectors is to reframe p-adic Hodge theory in terms of Witt vectors of characteristic 0 rings, and ultimately to globalize the constructions with an eye towards study of global ´etalecohomology, K-theory, and L-functions. We will pursue these goals in subsequent papers. 1. Witt vectors Throughout this section, let R denote an arbitrary commutative ring. For more details on the construction of p-typical Witt vectors, see [7, Section 0.1] or [5, Section 17]; the latter treats big Witt vectors as well as p-typical Witt vectors. Definition 1.1. For each nonnegative integer n, the ring Wpn (R) is defined to n+1 have underlying set Wpn (R) := R with an exotic ring structure characterized by functoriality in R and the property that for i = 0; : : : ; n, the pi-th ghost component map wpi : Wpn (R) ! R defined by pi pi−1 i wpi (r1; rp; : : : ; rpn ) = r1 + prp + ··· + p rpi is a ring homomorphism. These rings carry Frobenius homomorphisms F : Wpn+1 (R) ! Wpn (R); again functorial in R, such that for i = 0; : : : ; n, we have wpi ◦F = wpi+1 . Moreover, there are additive Verschiebung maps V : Wpn (R) ! Wpn+1 (R) defined by the formula V (r1; : : : ; rpn ) = (0; r1; : : : ; rpn ). There is a natural restriction map Wpn+1 (R) ! Wpn (R) obtained by forgetting the last component; define W (R) to be the inverse limit of the Wpn (R) via these restriction maps. The Frobenius homomorphisms at finite levels then collate to de- fine another Frobenius homomorphism F : W (R) ! W (R); there is also a collated Verschiebung map V : W (R) ! W (R). The ghost component maps also collate to define a ghost map: w : W (R) ! RN. We equip the target with component-wise ring operations; the map w is then a ring homomorphism. In either Wpn (R) or W (R), an element of the form (r; 0; 0;::: ) is called a Teichm¨uller element and denoted [r]. These elements are multiplicative: for all r1; r2 2 R,[r1r2] = [r1][r2]. We will need the following properties of Witt vectors for R arbitrary (not neces- sarily of characteristic p). See [6, Lemma 1.5]. Here and in what follows, we write x for a Witt vector with components x1; xp;::: . (a) For r 2 R, F ([r]) = [rp]. (b) For x 2 Wpn (R), (F ◦ V )(x) = px. (c) For x 2 Wpn (R) and y 2 Wpn+1 (R), V (xF (y)) = V (x)y. Pn i (d) For x 2 Wpn (R), x = i=0 V ([xpi ]). Remark 1.2. A standard method of proving identities about Witt vectors and their operations is reduction to the universal case: take R to be a polynomial ring in many variables over Z, form Witt vectors whose components are distinct variables, then verify the desired identities at the level of ghost components. This suffices because R is now p-torsion-free, so the ghost map is injective. We will need a couple of other p-divisibility properties. We first prove the fol- lowing lemma. ON THE WITT VECTOR FROBENIUS 3 Lemma 1.3. In W (Z=p2Z), we have p = (p; (−1)p−1; 0; 0;::: ) = [p] + V ([(−1)p−1]): p p−1 Proof. Write p = (x1; xp;::: ) 2 W (Z). Then x1 = p and xp = (p−p )=p = 1−p , which is congruent to 1 mod p2 if p > 2 and to 3 mod 4 if p = 2. We now show by 2 induction on n that for each n ≥ 1, we have xpi ≡ 0 mod p for 2 ≤ i ≤ n. The base case n = 1 is vacuously true. For the induction step, considering the pn-th ghost component of p, write n−1 n pn pn−1 X i pn−i p xpn = −x1 + p(1 − xp ) − p xpi : i=2 To complete the induction, it suffices to check that each term on the right side has p-adic valuation at least n + 2. This is clear for the first term because pn ≥ n + 2. p−1 p−1 p 3 For the second term we have xp ≡ (−1) mod p and so xp ≡ 1 mod p (we pn−1 3+n−2 treat p = 2 and p > 2 separately). We then have xp ≡ 1 mod p , so the second term is indeed divisible by pn+2. For the terms in the sum, the claim is n−i again clear because i + 2p ≥ i + 2(n − i + 1) ≥ n + 2. Lemma 1.4. Take x; y 2 W (R) with F (x) = y. p (a) For each nonnegative integer i, we have ypi = xpi +pxpi+1 +pfpi (x1; : : : ; xpi ) where fpi is a certain universal polynomial with coefficients in Z which is i+1 j homogeneous of degree p for the weighting in which xpj has weight p . pi+1 (b) For i ≥ 1, the coefficient of x1 in fpi equals 0. pi (c) For i ≥ 2, the coefficient of xp in fpi is divisible by p. p p−2 (d) The coefficient of xp in fp equals −p modulo p. (e) For p = 2 and i ≥ 2, f2i belongs to the ideal generated by the elements 2 2; x1; x2 − x4; x8; : : : ; x2i : Proof. By reduction to the universal case, we see that ypi equals a universal polyno- i+1 mial in x1; : : : ; xpi+1 with coefficients in Z which is homogeneous of degree p for p the given weighting. This polynomial is congruent to xpi modulo p by [7, (1.3.5)]. Each of the remaining assertions concerns a particular coefficient of this polynomial, and so may be checked after setting all other variables to 0. To finish checking (a), we must check that fpi does not depend on xpi+1 . We assume x1 = ··· = xpi = 0, i+1 i so that x = V ([xpi+1 ]); then F (x) = pV ([xpi+1 ]) = (0;:::; 0; pxpi+1 ). To check (b), we may assume that xp = xp2 = ··· = xpi = 0, so that x = [x1]. p In this case, F (x) = [x1], so the claim follows. To check (c), we may assume that x1 = xp2 = xp3 = ··· = xpi = 0, so that x = V ([xp]). In this case, the claim is 2 that ypi ≡ 0 mod p for i ≥ 2. Since F (x) = p[xp], by homogeneity it is sufficient to check the claim for xp = 1. In this case, it follows from Lemma 1.3.
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